Vedic maths › Squaring with the duplex

Squaring any number with the duplex

Squaring is multiplying a number by itself, so vertically and crosswise works. Because both numbers are the same, every crosswise pair comes up twice, and Vedic maths saves the work with the duplex (dvandva yoga): double one product instead of adding two equal ones.

The duplex of one digit is its square: 4 gives 16. Of two digits, twice their product: 43 gives 2 × 4 × 3 = 24. Of three digits, twice the outer two’s product plus the middle one’s square: 312 gives 2 × 3 × 2 + 1² = 13.

The method

1. Take the digits from the left in growing groups: the first, the first two, the first three… up to the whole number.
2. Then drop digits from the left until only the last is left.
3. Write the duplex of each group. Those are the parts of the answer.
4. Carry from the right, keeping one digit in each part.

Worked examples

Why it works

(10a + b)² = 100a² + 10 × 2ab + b², so the parts are a², 2ab and b²: the duplexes of a, ab and b.

With three digits, (100a + 10b + c)² = 10,000a² + 1000 × 2ab + 100 × (2ac + b²) + 10 × 2bc + c². Each bracket is a duplex.

Watch out

• The parts get big, so keep them apart until the end and carry carefully.
• For numbers ending in 5 or close to a base, squares ending in 5 and squaring near a base are quicker.

Try these

Work each one out, then tap it to see the answer.

1. 23²

   4 | 12 | 9 → 529

2. 58²

   25 | 80 | 64 → 3364

3. 91²

   81 | 18 | 1 → 8281

4. 123²

   1 | 4 | 10 | 12 | 9 → 15129

5. 214²

   4 | 4 | 17 | 8 | 16 → 45796

Practise mental maths in the game: play Make Me Smarter, free, with no sign-up.

Source: This method is from Bharati Krishna Tirtha’s Vedic Mathematics (1965). The explanation, examples and questions here are our own.

More Vedic maths

• Squares ending in 5: 75²: 7 × 8 = 56, then 25: 5625.
• All from 9, the last from 10: 1000 − 357: 9 − 3, 9 − 5, 10 − 7: 643.
• Multiplying near 100: 97 × 96: 97 − 4 = 93, 3 × 4 = 12: 9312.
• Squaring near 100: 96²: 96 − 4 = 92, 4² = 16: 9216.
• Vertically and crosswise: 23 × 41: 2 × 4 | 2 × 1 + 3 × 4 | 3 × 1: 943.
• Multiplying by 9, 99, 999: 437 × 999: 436, then 999 − 436 = 563: 436563.
• Last digits adding to 10: 63 × 67: 6 × 7 = 42, 3 × 7 = 21: 4221.
• Multiplying by 11: 253 × 11: 2 | 2 + 5 | 5 + 3 | 3: 2783.
• Checking with digit sums: 437 × 26 = 11362? 5 × 8 → 4, and 11362 → 4: it passes.
• Dividing by 9: 2312 ÷ 9: 2, 2 + 3, 5 + 1: 256, remainder 6 + 2 = 8.

Where Vedic maths comes from, all 16 sutras, and our sources: Vedic maths, explained.